A partial regularity theorem is established for a particular class of weak solutions to the system partial derivative u/partial derivative t - div(K(u)del u) = sigma(u)V phi(2), div(sigma(u)del phi) = 0 on a bounded domain in R(N). Under our assumptions, sigma(u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.